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A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p.

%I A185091 #29 Sep 08 2022 08:45:55
%S A185091 1,1,1,3,1,3,1,3,5,7,1,3,1,3,5,7,1,3,1,3,5,7,1,3,5,7,17,11,1,3,1,3,5,
%T A185091 7,13,11,1,3,5,7,1,3,1,3,5,7,1,3,5,7,17,11,1,3,5,7,29,11,1,3,1,3,5,7,
%U A185091 13,11,1,3,5,7,1,3,1,3,5,7,13,11,1,3,5,7,1,3,5,7,17,11,1,3,5,7,29
%N A185091 The smallest positive noncomposite q such that 2n-1 = 2p+q for some positive noncomposite p.
%C A185091 It is a Goldbach conjecture variant that terms exist for 2n-1 >= 5.
%C A185091 Lemma: N=2n-1 is coprime to q=a(n) unless N=3q. Proof: Suppose N and q are not coprime; so we have N=2p+q=iq with i=/=1=/=q, so (i-1)q=2p; now since q=/=2 (because N is odd), then q=p and i=3. QED.
%C A185091 Empirically, N=3q only for N=9,21.
%D A185091 Emile Lemoine, L'intermédiaire des mathématiciens, 1 (1894), 179; ibid 3 (1896), 151.
%H A185091 Jason Kimberley, <a href="/A185091/b185091.txt">Table of n, a(n) for n = 2..10002</a> (corrected by Michel Marcus, Jan 19 2019)
%H A185091 Brian H. Mayoh, <a href="http://dx.doi.org/10.1007/BF01939548">On the second Goldbach conjecture</a>, BIT Numerical Mathematics 6 (1966) 1, 48-50
%H A185091 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lemoine%27s_conjecture">Lemoine's conjecture</a>
%H A185091 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%o A185091 (Magma) A185091 := func<n|exists(q){q:q in[1..N div 3 by 2]|(q eq 1 or IsPrime(q))and IsPrime((N-q)div 2)}select q else -1 where N is 2*n-1>;[A185091(n):n in [3..94]];
%Y A185091 Records in this sequence are in A002092 occurring at 2n-1 in A002091.
%K A185091 nonn,easy
%O A185091 2,4
%A A185091 _Jason Kimberley_ (with thanks to _Hugo Pfoertner_), Sep 05 2011